Product-free sets with high density

TL;DR

This paper constructs dense sets of integers with asymptotic density close to 1 that contain no solutions to the equation ab=c, exploring generalizations and providing a specific example with density over 1/2.

Contribution

It introduces the existence of highly dense product-free sets of integers and analyzes their properties, including explicit examples and generalizations.

Findings

Existence of sets with density arbitrarily close to 1 that are product-free.

Construction of a specific product-free set with density greater than 1/2.

Analysis of generalizations of product-free sets.

Abstract

We show that there are sets of integers with asymptotic density arbitrarily close to 1 in which there is no solution to the equation ab=c, with a,b,c in the set. We also consider some natural generalizations, as well as a specific numerical example of a product-free set of integers with asymptotic density greater than 1/2.